straus introtoposttonaltheory

Introduction to Post-Tonal Theory

third edition

Joseph N. Straus Graduate Center

City University of New York

PEARSON

----Prentice Hall

Upper Saddle River, New Jersey 07458

Library of Congress Cataloging-in-Publication Data Straus, Joseph Nathan.

Introduction to post-tonal theory I Joseph N. Straus.-3rd ed. p. em.

ISBN 0-13-189890-6 L Music theory. 2. Atonality. 3. Twelve-tone system. 4. Musical analysis. L Title.

MT40.S96 2005 781.2'67-dc22

Editor-in-Chief: Sarah Touborg Senior Acquisitions Editor: Christopher T. Johnson Editorial Assi

~------~,~,,.,--------,----

vi Contents

Analysis 3 119 Webern, Movements for String Quartet, Op. 5, No. 4 Berg, "Schlafend triigt man mich;' from Four Songs, Op. 2

Chapter 4 130 Centricity, Referential Collections, and Triadic Post-Tonality

Tonality 130 Centricity 131 Inversional axis 133 The diatonic collection 140 The octatonic collection 144 The whole-tone collection 147 The hexatonic collection 149 Collectional interaction 150 Interval cycles 154 Triadic post-tonality 158 Exercises 170

Analysis 4 1 7 4 Stravinsky, Oedipus Rex, rehearsal nos. 167-70 Bart6k, Sonata, first movement

Chapter 5 Basic Twelve-Tone Operations

Twelve-tone series 182 Basic operations 183 Invariants 195 Exercises 201

Analysis 5 205 Schoenberg, Suite for Piano, Op. 25, Gavotte Stravinsky, In Memoriam Dylan Thomas

Chapter 6 More Twelve-Tone Topics

Subset structure 192

Webern and derivation 217 Schoenberg and hexachordal combinatoriality 222 Stravinsky and rotational arrays 231 Crawford and her "triple passacaglia" 234

182

217

Boulez and multiplication 235 Babbit and trichordal arrays 240 Exercises 246

Analysis 6 249 Webern, String Quartet, Op. 28, first movement Schoenberg, Piano Piece, Op. 33a

Appendix 1 List of Set Classes

Appendix 2 Index Vectors

Index

261

265

269

Preface

Compared to tonal theory, now in its fourth century of development, post-tonal the-ory is in its infancy. But in the past three decades, it has shown itselfto be an infant of prodigious growth and surprising power. A broad consensus has emerged among music theorists regarding the basic musical elements of post-tonal music-pitch, interval, motive, harmony, collection-and this book reports that consensus to a gen-eral audience of musicians and students of music. Like books on scales, triads, and simple harmonic progressions in tonal music, this book introduces basic theoretical concepts for the post-tonal music of the twentieth and twenty-first centuries.

Beyond basic concepts, the third edition of this book also contains information on many of the most recent developments in post -tonal theory, including expanded or new coverage of the following topics:

Transformational networks and graphs Contour theory Composing-out Atonal voice leading Atonal pitch space Triadic post-tonality (including voice-leading parsimony) Inversional symmetry and inversional axes Interval cycles Diatonic, whole-tone, octatonic, and hexatonic collections

As a result, this book is not only a primer of basic concepts but also an introduction to the current state of post-tonal theory, with its rich array of theoretical concepts and analytical tools.

Although this book can make no pretense to comprehensiveness either, either chronologically or theoretically-there is just too much great music and fascinating theory out there-this third edition explores a much wider range of composers and

vii

viii Preface

musical styles than its predecessors. Although the "classical" prewar repertoire of music by Schoenberg, Stravinsky, Bartok, Webern, and Berg still comprises the musical core, theoretical concepts are now also illustrated with music by Adams, Babbitt, Berio, Boulez, Britten, Cage, Carter, Cowell, Crawford, Crumb, Debussy, Feldman, Glass, Gubaidulina, Ives, Ligeti, Messiaen, Musgrave, Reich, Ruggles, Sessions, Shostakovich, Stockhausen, Varese, Wolpe, Wuorinen, and Zwilich.

As with the previous editions of this book, I received invaluable advice from many friends and colleagues based on their teaching experience. I am grateful to Wayne Alpern, Jonathan Bernard, Claire Boge, Ricardo Bordini, Scott Brickman, Michael Buchler, Uri Burstein, James Carr, Patrick Fairfield, Michael Friedmann, Edward Gollin, Dave Headlam, Gary Karpinski, Rosemary Killam, Bruce Quaglia, Daniel Mathers, Carolyn Mullin, Catherine Nolan, Jay Rahn, Nancy Rogers, Steven Rosenhaus, Art Samplaski, Paul Sheehan, Stephen Slottow, David Smyth, Harvey Stokes, Dmitri Tymoczko, and Joyce Yip. My thanks go also to Chris Johnson and Laura Lawrie at Prentice Hall for their expert editorial work at every stage. Michael Berry provided additional editorial assistance. Closer to home, in matters both tangi-ble and intangible, Sally Goldfarb has offered continuing guidance and support beyond my ability to describe or repay. Adam and Michael helped, too.

Joseph N. Straus Graduate Center

City University of New York

Preface ix

The publishers are gratefully acknowledged for permission to use the following musical examples: John Adams. HARMONIUM Words by Emily Dickinson. Copyright 1981 by Associated Music Publishers, Inc. (BMI) International Copyright Secured. All Rights Reserved. Reprinted by Permission. Milton Babbitt, STRONG QUARTET NO. 2 Copyright 1967 (Renewed) by Associated Music Publishers. Inc. (BMI). International Copyright Secured. All rights Reserved. Reprinted by Permission. Babbitt, SEMI-SIMPLE VARIATIONS. 1957 Theodore Presser Company. Used by pem1ission of the Publisher. Music of Bela Bartok PIANO SONATA NO. I Copyright 1927 by Boosey & Hawkes, Inc. for the USA. Copyright Renewed. Reprinted by permission. STRING QUARTET NO. 4 Copyright 1929 by Boosey & Hawkes, Inc. for the USA. Copyright Renewed. Reprinted by permission. SONATA FOR TWO PIANOS AND PERCUSSION Copyright 1942 by Hawkes & Son (London) Ltd. Copyright Renewed. Revised version Copyright 1970 by Hawkes & Son (London) Ltd. Copyright Renewed. Reprinted by permission ofBoosey & Hawkes, Inc. Alban Berg, Violin Concerto 1936 by Universal Edition A. G., Vienna. Renewed. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. and Canadian agent for Universal Edition A.G., Vienna. Pierre Boulez, LE MARTEAU SANS MAITRE 1954, 1957 by Universal Edition Ltd., London. Poemes de Rene Char: 1954 by Jose Corti Editur, Paris. All Rights Reserved. Used by pennission of European American Music Distributors Corporation, sole U.S. and Canadian agent for Universal Edition Ltd., London. Pierre Boulez, STRUCTURES lA 1955 by Universal Edition Ltd., London. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. and Canadian agent for Universal Edition Ltd., London. John Cage, FOR PAUL TAYLOR AND ANITA DENECKS, 1978 by Henmar Press Inc., New York. All rights reserved. Used by permission. Elliott Carter, STRING QUARTET NO. 2 Copyright 1961 (Renewed) by Associated Music Publishers, Inc. (BMI) International Copyright Secured. All Rights Reserved. Reprinted by Permission. Henry Cowell, THE BANSHEE Copyright 1930 (Renewed) by Associated Music Publishers, Inc. (BMI). International Copyright Secured. All Rights Reserved. Reprinted by Permission. Crawford (Seeger), DIAPHONIC SUITE NO. I Copyright 1972 Continuo Music Press, Inc. International Copyright Secured. Made in U.S.A. All rights reserved. Reprinted by permission. Crawford (Seeger), STRING QUARTET. Merion Music, Inc. Used by permission. Sofia Gubaidulina, STRING TRIO. Copyright Chante Du Monde. Sub-published in the United States, Canada and Mexico by G. Schirmer, Inc. (ASCAP) International Copyright Secured. All Rights Reserved. Reprinted by Permission. George Crumb, ANCIENT VOICES OF CHILDREN, Copyright 1971 by C. F. Peters Corporation Inc., New York. All Rights Reserved. Used by permission. George Crumb, MAKROKOSMOS, VOL I, No. 1, Copyright 1973 by C. F. Peters Corporation, New York. All Rights Reserved. Used by permission. George Crumb, MAKROKOSMOS, VOL 2, No. 8 ("Gargoyles"), Copyright 1973 by C. F. Peters Corporation, New York. All Rights Reserved. Used by permission. Morton Feldman, DURATIONS III, No. 3, Copyright 1962 by C. F. Peters Corporation, New York. All Rights Reserved. Used by permission. Morton Feldman, PROJECTION No. I FOR SOLO CELLO, Copyright 1962 by C. F. Peters Corporation, New York. All Rights Reserved. Used by permission. Ligeti, TEN PIECES FOR WIND QUINTET, NO.9, Copyright 1969 by Schott Musik InternationaL Renewed. All Rights Reserved. Used by permission of European American Music Distributors LLC. sole US and Canadian agent for Schott Musik International.

X Preface

Messiaen, QUARTET FOR THE END OF TIME. 1942 Durand S.A. Used by permission. Sole Agent U.S.A .. Theodore Presser Co. MARY, QUEEN OF SCOTS An Opera in 3 Acts based on the play "Morey" by Amalia Elguera. Copyright 1977 by Novello & Company Limited. International copyright Secured. All Rights Reserved. Reprinted by permission. Schoenberg, THE BOOK OF THE HANGING GARDENS, No. II. Used by permission of Belmont Music Publishers, Los Angeles, California 90049 Copyright 1914 by Universal Edition. Copyright renewed 1941 by Arnold Schoenberg. Schoenberg, FIVE PIECES FOR ORCHESTRA, "FARBEN." Arranged for two pianos by Anton Webern. 1913 by C.F. Peters. Used by permission. Schoenberg, LITTLE PIANO PIECES. Op. 19, No. 2. Used by permission of Belmont Music Publishers, Los Angeles, Califomia 90049. Copyright 1913 by Universal Edition. Copyright renewed 1940 by Arnold Schoenberg. Schoenberg, PIANO PIECES, Op. II No. I. Used by permission of Belmont Music Publishers, Los Angeles, Califomia 90049. Copyright 1910 by Universal Edition. Copyright renewed 1938 by Arnold Schoenberg. Schoenberg, PIANO PIECES, Op. 33a. Used by permission of Belmont Music Publishers, Los Angeles, California 90049. Copyright 1929 by Universal Edition. Copyright renewed 1956 by Gertrud Schoenberg. Schoenberg, PIERROT LUNAIRE, "NACHT." Used by permission of Belmont Music Publishers, Los Angeles, California 90049. Copyright 1914 by Universal Edition. Copyright renewed 1941 by Arnold Schoenberg. Schoenberg, STRING QUARTET NO. 3. Used by permission of Belmont Music Publishers, Los Angeles, Califomia 90049. Copyright 1927 by Universal Edition. Copyright renewed 1954 by Gertrud Schoenberg. Schoenberg, STRING QUARTET NO. 4, Op. 37. Copyright 1939 (Renewed) by G. Schirmer, Inc. {ASCAP). International Copyright Secured. All Rights Reserved. Reprinted by Permission. Schoenberg, SUITE, Op. 25. Used by permission of Belmont Music Publishers, Los Angeles, California 90049. Copyright 1925 by Universal Edition. Copyright renewed 1952. Dmitri Shostakovich. STRING QUARTET NO.4 IN D MAJOR, OP. 83. Copyright 1954 (Renewed) by G. schirmer, Inc. (ASCAP). International Copyright Secured. All Rights Reserved. Reprinted by Permission. Dmitri Shostakovich, STRING QUARTET NO. 12 IN B FLAT MAJOR. OP. 133. Copyright 1945 (Renewed) by G. schirmer, Inc. (ASCAP). International Copyright Secured. All Rights Reserved. Reprinted by Permission. Stockhause, KLAVIERSTUCK 1-IV/NO. 2, Copyright 1954 by Universal Edition (London) Ltd., London. Renewed. All Rights Reserved. Used by permission of European American Music Distributors LLC, US and Canadian agent for Universal Edition (London) Ltd., London. Music of Igor Stravinsky Agon Copyright 1957 by Hawkes & Son (London), Ltd. Copyright Renewed. Reprinted by permission of Boosey & Hawkes, Inc. IN MEMORIAM DYLAN THOMAS Copyright 1954 by Hawkes & Son (London), Ltd. Copyright Renewed. Reprinted by permission of Boosey & Hawkes, Inc. LES NOCES Copyright 1922, 1990 by Chester Music Limited, 8/9 Frith Street, London WID 3JB, United Kingdom. International Copyright Secured. All Rights Reserved. Reprinted by Permission. OEDIPUS REX Copyright 1927 by Hawkes & Son (London), Ltd. Copyright Renewed. Revised ver-sion Copyright 1949, 1950 by Hawkes & Son (London), Ltd. Copyright Renewed. Reprinted by permis-sion ofBoosey & Hawkes, Inc. PETROUCHKA Copyright 1912 by Hawkes & Son (London), Ltd. Copyright Renewed. Revised ver-sion Copyright 1948 by Hawkes & Son (London), Ltd. Copyright Renewed. Reprinted by permission of Boosey & Hawkes, Inc.

Preface xi

THE RAKE'S PROGRESS Copyright 1951 by Hawkes & Son (London), Ltd. Copyright Renewed. Reprinted by permission of Boosey & Hawkes, Inc. SYMPHONY OF PSALMS Copyright 1931 by Hawkes & Son (London), Ltd. Copyright Renewed. Revised version Copyright 1948 by Hawkes & Son (London), Ltd. Copyright Renewed. Reprinted by permission of Boosey & Hawkes, Inc. SERENADE IN A Copyright 1926 by Hawkes & Son (London). Ltd. Copyright Renewed. Reprinted by permission of Boosey & Hawkes, Inc. A SERMON, A NARRATIVE. AND A PRAYER Copyright 1961 by Hawkes & Son (London), Ltd. Copyright Renewed. Reprinted by permission of Boosey & Hawkes, Inc. REQUIEM CANTICLES Copyright 1967 by Hawkes & Son (London). Ltd. Copyright Renewed. Reprinted by permission of Boosey & Hawkes, Inc. THREE PIECES FOR STRING QUARTET Copyright 1928 by Hawkes & Son (London), Ltd. Copyright Renewed. Reprinted by permission of Boosey & Hawkes, Inc. SYMPHONY INC Copyright 1948 by Schott Musik International. Renewed. All Rights Reserved. Used by permission of European American Music Distributors Corporation. sole U.S. and Canadian agent for Schott Musik International. Edgard Valese, DENSITY 21.5, 1946 G. Ricordi & Company (SIAE). All All rights for the world obo G. Ricordi & Company (SIAE) administered by BMG Ricordi S.P.A.-Milan (SIAE). All rights for the U.S. obo BMG Ricordi S.P.A.- Milan (SIAE) administered by Careers-BMG Music Publishing, Inc. (BMI). Edgard Valese, HYPERPRlSM, 1946 G. Ricordi & Company (SIAE). All All rights for the world obo G. Ricordi & Company (SIAE) administered by BMG Ricordi S.P.A.-Milan (SIAE). All rights for the U.S. obo BMG Ricordi S.P.A-Milan (SIAE) administered by Careers-BMG Music Publishing, Inc. (BMI). Anton Webern, 5 LIEDER, OP. 3, Used by permission of European American Music Distributors LLC, US and Canadian agent for Universal Edition. Anton Webem, MOVEMENTS FOR STRING QUARTET OP. 5 1922 by Universal Edition A.G., Vienna. renewed 1949 by Anton Webem's Erben.Ail Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. and Canadian agent for Universal Edition A.G., Vienna.

Anton Webem, STRING QUARTET, OP. 28 1955 by Universal Edition A.G., Vienna. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. and Canadian agent for Universal Edition A.G., Vienna. Anton Webern, THREE SONGS, OP. 25 1956 by Universal EdilionA.G., Vienna. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. and Canadian agent for Universal Edition A. G., Vienna. Anton Webern, VARIATIONS, OP. 27 1937 by Universal Edition A.G., Vienna. renewed. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. and Canadian agent for Universal Edition A.G., Vienna. Anton Webern, SONGS, OP. 14 1924 by Universal Edition A.G., Vienna. renewed 1952 by Anton Webem's Erben. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. and Canadian agent for Universal Edition A.G .. Vienna. Anton Webem. CONCERTO, OP. 24 1948 by Universal Edition A.G., Vienna. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. and Canadian agent for Universal Edition A. G., Vienna. Stefan Wolpe, "FORM FOR PIANO," MM. 1-4 Copyright 1962 by Tonos Musikverlags GmbH, Darmstadt/Germany. Charles Wuorinen, TWELVE SHORT PIECES. NO. 3, Copyright 1980 by C. F. Peters Corporation, New York. All Rights Reserved. Used by permission.

Chapter 1 Basic Concepts and Definitions

Octave Equivalence There is something special about the octave. Pitches separated by one or more octaves are usually perceived as in some sense equivalent. Our musical notation reflects that equivalence by giving the same name to octave-related pitches. The name A, for example, is given not only to some particular pitch, like the A a minor third below middle C, but also to all the other pitches one or more octaves above or below it. Octave-related pitches are called by the same name because they sound so much alike and because Western music usually treats them as functionally equivalent.

Equivalence is not the same thing as identity. Example l-l shows a melody from Schoenberg's String Quartet No. 4, first as it occurs at the beginning of the first movement and then as it occurs a few measures from the end.

a.

Example l-1 Two equivalent melodies (Schoenberg, String Quartet No.4).

The two versions are different in many ways, particularly in their rhythm and range. The range of the second version is so wide that the first violin cannot reach all of the

1

2 Basic Concepts and Definitions notes; the cello has to step in to help. At the same time, however, it is easy to recog-nize that they are basically the same melody-in other words, that they are octave equivalent.

In Example 1-2, the opening of Schoenberg's Piano Piece, Op. 11, No.1, com-pare the first three notes of the melody with the sustained notes in measures 4-5.

Piano

Example 1-2 Two equivalent musical ideas (Schoenberg, Piano Piece, Op. II, No. l ).

There are many differences between the two collections of notes (register, articula-tion, rhythm, etc.), but a basic equivalence also. They are equivalent because they both contain a B, a OJ, and a G.

We find the same situation in the passage shown in Example 1-3, from a string quartet movement by Webem. The first three notes of the viola melody--G. B, and 0-retum as the cadential chord at the end of the phrase. The melody and the chord are octave equivalent.

Example l-3 Two equivalent musical ideas (Webem, Movements for String Quartet, Op. 5, No. 2).

When we assert octave equivalence, and other equivalences we will discuss later, our object is not to smooth out or dismiss the variety of the musical surface. Rather, we seek to discover the relationships that underlie the surface and lend unity and coherence to musical works.

Basic Concepts and Definitions 3

Pitch Class We will distinguish between a pitch (a tone with a certain frequency) and a pitch class (a group of pitches with the same name). Pitch-class A, for example, contains all the pitches named A. To put it the other way around, any pitch named A is a mem-ber of pitch-class A. Sometimes we will speak about specific pitches; at other times we will talk, more abstractly, about pitch classes. When we say that the lowest note on the cello is a C, we are referring to a specific pitch. We can notate that pitch on the second ledger line beneath the bass staff. When we say that the tonic of Beethoven's Fifth Symphony is C, we are referring not to some particular pitch C, but to pitch-class C. Pitch-class C is an abstraction and cannot be adequately notated on musical staves. Sometimes, for convenience, we will represent a pitch class using musical notation. In reality, however, a pitch class is not a single thing; it is a class of things, of pitches one or more octaves apart.

The passage shown in Example 1-4 consists of seventeen three-note chords. The pitches change as the instruments jump around, but each chord contains the same three pitch classes: H, G, and A~ (notice that the violin is playing harmonics that pro-duce a pitch two octaves higher than the filled-in notehead).

Slow ~~~ ~..,_ ~..,_ f\ II"'- -"- II"'- .. .. --"'--=::k Violin

..,

L L L L Tuba

#ii' # # #ii' #ii' ;; ~~

....

l~ ~- ~- b ... ~ ~ 15~/M .. == :;;;. = :;;;. " -~ Piano

.., ;; w w- w

Example 1-4 Many pitches, but only three pitch classes: Fl, G, and Ab (Feldman, Durations Ill, No.3).

Enharmonic Equivalence In common-practice tonal music, a Bb is not the same as an M. Even on an equal-tempered instrument like the piano, the tonal system gives B~ and AI different func-tions and different meanings, representing different degrees of the scale. In G major, for example, M is ;i whereas m is ~3. and scale-degrees 2 and 3 have very different musical roles both melodically and harmonically. These distinctions are largely

4 Basic Concepts and Definitions abandoned in post-tonal music, however, where notes that are enharmonically equiv-alent (like B~ and AI) are also functionally equivalent. For example, the passage in Example 1-5 involves three repetitions: the A returns an octave higher, the B returns

two octaves lower, and the A~ returns three octaves higher as a Gl. A~ and GM are enharmonically equivalent.

f\ .-3-, ;-3~ r-5~:;;. ~

=

I -v 4 L I b 5 8P p LJ p 1iif Sf ~J J : f

Example 1-5 Enharmonic equivalence (Stockhausen, Klavierstuck Ill).

There may be isolated moments where a composer notates a pitch in what seems like a functional way (sharps for ascending motion and flats for descending, for example). For the most part, however, the notation is functionally arbitrary, deter-mined by simple convenience and legibility. The melodies in Example 1-6 are enhar-monically equivalent (although the first one is much easier to read).

a.

b.

Example 1-6 Enharmonic equivalence.

Integer Notation Octave equivalence and enharmonic equivalence leave us with only twelve different pitch classes. All the Bls, Os, and ms are members of a single pitch class, as are all the Os and Dbs, all the Cs, Ds, and FJJ,s, and so on. We will often use integers from 0 through 11 to refer to the pitch classes. Figure 1-l shows the twelve different pitch classes and some of the contents of each.

Basic Concepts and Definitions integer name

0

2 3 4 5 6 7 8 9

10 II

pitch-class content B~. C. Dii>

Ci,Db

Figure 1-1

C., D, FJJ, Di,B

D.,E,R EI,F,GI:h H,a

G,AI:h GI,M

G.,A,Bit. Ai,m

A",B,G

5

We will use a "fixed do" notation: the pitch class containing the Cs is arbitrarily assigned the integer 0 and the rest follows from there.

Integers are traditional in music (figured-bass numbers, for example) and use-ful for representing certain musical relationships. We will never do things to the inte-gers that don't have musical significance. We won't divide integers, because, while dividing 7 into 11 makes numerical sense, dividing G into B doesn't make much musical sense. Other arithmetical operations, however, will prove musically usefuL We will, for example, subtract numbers, because, as we will see, subtraction gives us a simple way of talking about intervals. Computing the distance between 7 and II by subtracting 7 from II makes numerical sense, and the idea of computing the distance between G and B makes musical sense. We will use numbers and arithmetic to model interesting aspects of the music we study. The music itself is not ''mathematical" any more than our lives are "mathematical" just because we count our ages in integers. In this book. we will identify pitch classes with either traditional letter notation or inte-gers, whichever seems clearest and easiest in a particular context.

Mod 12 Every pitch belongs to one of the twelve pitch classes. Going up an octave (adding twelve semitones) or going down an octave (subtracting twelve semi tones) will just produce another member of the same pitch class. For example, if we start on the B above middle C (a member of pitch class 3) and go up twelve semitones, we end up back on pitch class 3. In other words, in the world of pitch classes, 3 + 12 = 15 = 3. More generally, any number larger than I1 or smaller than 0 is equivalent to some integer from 0 to 11 inclusive. To figure out which one, just add or subtract 12 (or any multiple of 12). Twelve is called the modulus, and our theoretical system fre-quently will rely upon arithmetic modulo 12, for which mod 12 is an abbreviation. In a mod 12 system, -12 = 0 = 12 24, and so on. Similarly, -13, 23, and 35 are all

6 Basic Concepts and Definitions

equivalent_to 11 (and to each other) because they are related to II (and to each other) by adding or subtracting 12.

It is easiest to understand these (and other) mod 12 relationships by envision-ing a circular clockface, like the one in Figure 1-2.

11 0

lO 2

9 3

8 4 7 6 5

Figure 1-2

In a mod 12 system, moving 12 (or a multiple of 12) in either direction only brings you back to your starting point. As a result, we will generally be dealing only with integers between 0 and 11 inclusive. When we are confronted with a number larger than 11 or smaller than 0, we will usually write it, by adding or subtracting 12, as an integer between 0 and 11. We will sometimes use negative numbers (for example, when we want to suggest the idea of descending), and we will sometimes use num-bers larger than 11 (for example, when discussing the distance between two widely separated pitches), but in general we will discuss such numbers in terms of their mod 12 equivalents.

We locate pitches in an extended pitch space, ranging in equal-tempered semi-tones from the lowest to the highest audible tone. We locate pitch classes in a modu-lar pitch-class space, as in Figure 1-2, which circles back on itself and contains only the twelve pitch classes. It's like the hours of the day or the days of the week. As our lives unfold in time, each hour and each day are uniquely located in linear time, never to be repeated. But we can be sure that, if it's eleven o'clock now, it will be eleven o'clock again in twelve hours (that's a mod 12 system), and that if it's Friday today, it will be Friday again in seven days (that's a mod 7 system). Just as our lives unfold simultaneously in linear and modular time, music unfolds simultaneously in pitch and pitch-class space.

Intervals

Because of enharmonic equivalence, we will no longer need different names for intervals with the same absolute size-for example, diminished fourths and major thirds. In tonal music, such distinctions are crucial; intervals are defined and named according to their tonal function. A third, for example, is an interval that spans three steps of the diatonic scale, while a fourth spans four steps. A major third is conso-nant, while a diminished fourth is dissonant. In music that doesn't use diatonic scales


and doesn't systematically distinguish between consonance and dissonance, it seems cumbersome and even misleading to use traditional interval names. It will be easier and more accurate musically just to name intervals according to the number of semi-tones they contain. The intervals between C and E and between C and P, both contain four semitones and are both instances of interval 4, as are BI-P,, C-Dx, and so on.

Example 1-7 extracts a series of seven harmonic intervals played in rhythmic unison by the second violin and viola in a passage from Elliott Carter's String Quartet No. 3, a piece in which two instrumental duos often play distinct intervals. The first six intervals are spelled as major thirds while the seventh is spelled as a diminished fourth, but in this musical context it is clear that all seven intervals are to be understood as enharmonically equivalent.

gva _______ ___ ------- ~ __ ~~ ----, 2ndViolin ._ ~;

8""- -------------------.-'-- '

Example l-7 Enharmonically equivalent intervals (Carter, String Quartet No. 3, mm. 245-62).

Figure 1-3 gives some traditional interval names and the number of semi tones they contain.

traditional name no. of traditional name no. of semitones semitones

unison 0 major 6th, diminished 7th 9 minor 2nd I augmented 6th, minor 7th 10 major 2nd, diminished 3rd 2 major 7th II minor 3rd, augmented 2nd 3 octave 12 major 3rd, diminished 4th 4 minor 9th l3 augmented 3rd, perfect 4th 5 major 9th 14 augmented 4th, diminished 5th 6 minor lOth 15 perfect 5th, diminished 6th 7 major lOth 16 augmented 5th, minor 6th 8

Figure l-3


Pitch Intervals A pitch interval is simply the distance between two pitches, measured by the number of semitones between them. A pitch interval, which will be abbreviated ip, is created when we move from pitch to pitch in pitch space. It can be as large as the range of our hearing or as small as a semitone. Sometimes we will be concerned about the direc-tion of the interval, whether ascending or descending. In that case, the number will be preceded by either a plus sign (to indicate an ascending interval) or a minus sign (to indicate a descending interval). Intervals with a plus or minus sign are called directed or ordered intef11als. At other times, we will be concerned only with the absolute space between two pitches. For such unordered intef11als, we will just provide the number of semitones between the pitches.

Whether we consider the interval ordered or unordered depends on our particu-lar analytical interests at the time. Example 1-8 shows the opening melody from Schoenberg's String Quartet No. 3, and identifies both its ordered and unordered pitch intervals.

ordered pitch intervals: -1

unordered pitch intervals:

+3 -5 -6

3 5 6

+15

15 6 4

Example 1-8 Ordered and unordered pitch intervals (Schoenberg, String Quartet No. 3).

The ordered pitch intervals focus attention on the contour of the line, its balance of rising and falling motion. The unordered pitch intervals ignore contour and concen-trate entirely on the spaces between the pitches.

Ordered Pitch-Class Intervals A pitch-class interval is the distance between two pitch classes. A pitch-class inter-val, which will be abbreviated i, is created when we move from pitch class to pitch class in modular pitch-class space. It can never be larger than eleven semitones. As with pitch intervals, we will sometimes be concerned with ordered intervals and sometimes with unordered intervals. To calculate pitch-class intervals, it is best to think again of a circular clockface as in Figure 1-2. We will consider clockwise movement to be equivalent to movement upward, and counterclockwise movement equivalent to movement downward. With this in mind, the ordered interval from 0 to A, for example, is -4 or +8. In other words, from pitch-class 0, one can go either up eight semitones or down four semitones to get to pitch-class A. This is because +8 and -4 are equivalent (mod 12). It would be equally accurate to call that interval 8 or


-4. By convention, however, we will usually denote ordered pitch-class intervals by an integer from 0 to 11. To state this as a formula, we can say that the ordered interval from pitch class x to pitch classy is y- x (mod 12). Notice that the ordered pitch class interval from A to C# (1- 9 -8 (mod 12) = 4) is different from that from C# to A (8), since, when discussing ordered pitch-class intervals, order matters. Four and 8 are each other's complement mod 12, because they add up to 12, as do 0 and 12, I and 11, 2 and 10,3 and 9, and 5 and 7. Six is its own complement mod 12.

Figure l-4 calculates some ordered pitch-class intervals using the formula.

The ordered pitch-class interval from 0 to B is 3 - I = 2 from Ei>to 0 is I 3 = lO from B to F is 5 - II = 6 from D to m is lO - 2 8 from m to C# is I lO = 3

Figure l-4

You will probably find it faster just to envision a musical staff, keyboard, or a clock-face. To find the ordered pitch-class interval between 0 and A, just envision the C# and then count the number of half-steps you will need to go upward (if you are envi-sioning a staff or keyboard) or clockwise (if you are envisioning a clockface) to the nearest A.

Unordered Pitch-Class Intervals For unordered pitch-class intervals, it no longer matters whether you count upward or downward. All we care about is the space between two pitch classes. Just count from one pitch class to the other by the shortest available route, either up or down. The for-mula for an unordered pitch-class interval is x - y (mod 12) or y - x (mod 12), whichever is smaller. The unordered pitch-class interval between C# and A is 4, because 4 (1 - 9 = -8 = 4) is smaller than 8 (9 I = 8). Notice that the unordered pitch-class interval between C# and A is the same as that between A and Cl. It is 4 in both cases, since from A to the nearest Cl is 4 and from Cl to the nearest A also is 4. Including the unison, 0, there are only seven different unordered pitch-class intervals, because, to get from one pitch class to any other, one never has to travel farther than six semitones. Figure 1-5 calculates some unordered pitch-class intervals using the formula. The correct answer is underlined.

The unordered pitch-class interval between Cl and B is 3 - I = 2 or I - 3 = I 0 B and Cl is I 3 = lO or 3 - I = 2 B and F is 5 II = .6 or 11 - 5 = .6 D and Bhs I 0 2 = 8 or 2 - lO = 1 Bb and 0 is I - lO = .3. or lO - I = 9

Figure l-5


Again, you will probably find it faster just to envision a clockface, musical staff, or keyboard. To find the unordered pitch-class interval between m and H, for example, just envision a B~ and count the number of semi tones to the nearest available H ( 4 ).

In Example l-9a (again the opening melody from Schoenberg's String Quartet No.3), the first interval is ordered pitch-class interval 11, to be abbreviated as i 11.

a. ordered pltch~class intervals: 1 1

b. ordered pitch-class intervals:

unordered pitch-class intervals: 4 4

Example 1-9 Ordered and unordered pitch-class intervals (Schoenberg, String Quartet No. 3).

)'hat's because to move from B tom one moves -1 or its mod 12 equivalent, ill. Eleven is the name for descending semitones or ascending major sevenths or their compounds. If them had come before the B, the interval would have been i1, which is the name for ascending semitones or descending major sevenths or their com-pounds. And that is the interval described by the two subsequent melodic gestures, 0-D and F-H. As ordered pitch-class intervals, the first is different from the second and third. As unordered pitch-class intervals, all three are equivalent. In Example 1-9b, two statements of i4 are balanced by a concluding i8; all three represent unordered pitch-class interval4.

Interval Class An unordered pitch-class interval is also called an interval class. Just as each pitch-class contains many individual pitches, so each interval class contains many individ-ual pitch intervals. Because of octave equivalence, compound intervals-intervals larger than an octave-are considered equivalent to their counterparts within the octave. Furthermore, pitch-class intervals larger than six are considered equivalent to their complements mod 12 (0 = 12, 1 = 11, 2 = 10, 3 = 9, 4 = 8, 5 = 7, 6 = 6). Thus, for


example, intervals 23, 13, 11, and 1 are all members of interval class 1. Figure 1-6 shows the seven different interval classes and some of the contents of each.

--,--..

al class 0 I 2 3 4 5

intervals 0,12,24 1,11,13 2,10,14 3,9,15 4,8,16 5,7,

Figure 1-6

We thus have four different ways of talking about intervals: ordered pitch inter-val, unordered pitch interval, ordered pitch-class interval, and unordered pitch-class interval. If in some piece we come across the musical figure shown in Example l-10, we can describe it in four different ways.

ordered pitch interval: + 19 unordered pitch interval: 19

ordered pitch-class interval: 7 unordered pitch-class interval: 5

Example 1-10 Four ways of describing an interval.

If we call it a+ 19, we have described it very specifically, conveying both the size of the interval and its direction. If we call it a 19, we express only its size. If we call it a 7, we have reduced a compound interval to its within-octave equivalent. If we call it a 5, we have expressed the interval in its simplest, most abstract way. None of these labels is better or more right than the others-it's just that some are more concrete and specific while others are more general and abstract. Which one we use will depend on what musical relationship we are trying to describe.

It's like describing any object in the world-what you see depends upon where you stand. If you stand a few inches away from a painting, for example, you may be aware of the subtlest details, right down to the individual brushstrokes. If you stand back a bit, you will be better able to see the larger shapes and the overall design. There is no single "right" place to stand. To appreciate the painting fully, you have to be willing to move from place to place. One of the specially nice things about music is that you can hear a single object like an interval in many different ways at once. Our different ways of talking about intervals will give us the flexibility to describe many different kinds of musical relationships.

Interval-Class Content The quality of a sonority can be roughly summarized by listing all the intervals it contains. To keep things simple, we will generally take into account only interval classes (unordered pitch-class intervals). The number of interval classes a sonority contains depends on the number of distinct pitch classes in the sonority. The more


pitch classes, the greater the number of interval classes. Figure 1-7 summarizes the number of interval classes in sonorities of all sizes. (We won't bother including the occurrences of interval class 0, which will always be equal to the number of pitch classes in the sonority.)

no. of pitch classes I 2 3 4 5 6 7 8 9

10 11 12

no. of interval classes 0 l 3 6

10 15 21 28 36 45 55 66

Figure 1-7 For any given sonority, we can summarize the interval content in scoreboard

fashion by indicating, in the appropriate column, the number of occurrences of each of the six interval classes, again leaving out the occurrences of interval class 0. Such a scoreboard conveys the essential sound of a sonority. Notice that now we are count-ing all of the intervals in the sonority, not just those formed by notes that are right next to each other. That is because all of the intervals contribute to the overall sound.

Example I-ll refers to the same passage and the same three-note sonority dis-cussed back in Example l-2.

Piano

o;;~al class ~f occurrences

Example 1-11 Interval-class content of a three-note motive (Schoenberg, Piano Piece, Op. ll, No. 1).


Like any three-note sonority, it contains three intervals, in this case one occurrence each of interval classes I, 3, and 4 (no 2s, 5s. or 6s). How different this is from the sonorities preferred by Stravinsky in the passage from his opera The Rake's Progress, shown in Example 1-12 or by Varese in the passage from his solo flute piece Density 21.5 shown in Example 1-13! Stravinsky's chords contain only 2s and 5s and Varese's melodic cells contain only Is, 5s, and 6s. The difference in their sound is a reflection of the difference in their interval content.

Example 1-12 Interval-class content of a three-note motive (Stravinsky, The Rake's Progress, Act I).

Example 1-13 Interval-class content of a three-note motive (Varese, Densi~v 21.5, mm. 11-14).


Interval-Class Vector Interval-class content is usually presented as a string of six numbers with no spaces intervening. This is called an interval-class vector. The first number in an interval-class vector gives the number of occurrences of interval class l; the second gives the number of occurrences of interval class 2; and so on. The interval-class vector for the sonority in Example l-11 is 101100, the interval-class vector for the sonor-ity in Example 1-12 is 010020, and the interval-class vector for the sonority in Example 1-13 is 100011.

We can construct a vector like this for sonorities of any size or shape. A tool like the interval-class vector would not be nearly so necessary for talking about tradi-tional tonal music. There, only a few basic sonorities-four kinds of triads and five kinds of seventh chords-are regularly in use. In post-tonal music, however, we will confront a huge variety of harmonies. The interval-class vector will give us a conve-nient way of summarizing their basic sound.

Even though the interval-class vector is not as necessary a tool for tonal music as for post-tonal music, it can offer an interesting perspective on traditional forma-tions. Example l-14 calculates the interval-class vector for the major scale.

1 213141516

l I b I 2

l 2 2

1 I 2

l 1 1

I I

I

of occurrences 2 5 4 3 6 l

Example l-14 Interval-class vector for the major scale.


Notice our methodical process of extracting each interval class. First, the intervals formed with the first note are extracted, then those formed with the second note, and so on. This ensures that we find all the intervals and don't overlook any. As with any seven-note collection, there are 21 intervals in all.

Certain intervallic properties of the major scale are immediately apparent from the interval-class vector. It has only one tritone (fewer than any other interval) and six occurrences of interval-class 5, which contains the perfect fourth and fifth (more than any other interval). This probably only confirms what we already knew about this scale, but the interval-class vector makes the same kind of information available about less familiar collections. The interval-class vector of the major scale has another interesting property-it contains a different number of occurrences of each of the interval classes. This is an extremely important and rare property (only three other collections have it) and it is one to which we will return. For now, the important thing is the idea of describing a sonority in terms of its interval-class content.

BIBLIOGRAPHY

The material presented in Chapter I (and in much of Chapters 2 and 3 as well) is also discussed in three widely used books: Allen Forte, The Structure of Atonal Music (New Haven: Yale University Press, 1973); John Rahn, Basic Atonal Theory (New York: Longman, 1980); and George Perle, Serial Composition and Atonality, 6th ed., rev. (Berkeley and Los Angeles: University of California Press, 1991). Two impor-tant books offer profound new perspectives on this basic material, and much else besides: David Lewin, Generalized Musical Intervals and Transfonnations (New Haven: Yale University Press, 1987); and Robert Morris, Composition with Pitch Classes (New Haven: Yale University Press, 1987). Ambitious students will be inter-ested in Robert Morris, Class Notes for Atonal Music Theory (Hanover, N.H.: Frog Peak Music, 1991) and Class Notes for Advanced Atonal Music Theory (Hanover, N.H.: Frog Peak Music, 2001). For an aural skills approach to post-tonal theory, see Michael Friedmann, Ear Training for Twentieth-Century Music (New Haven: Yale University Press, 1990).

Exercises

I.

THEORY

Integer Notation: Any pitch class can be represented by an integer. In the com-monly used "fixed do" notation, C = 0, 0 = 1, D = 2, and so on.

1. Represent the following melodies as strings of integers:

16

a.

c.

2.

Basic Concepts and Definitions

Show at least two ways each of the following strings of integers can be notated on a musical staff: a. 0 1 3 9 2 11 4 10 7 8 56 b. 24124676424212 c. 0 11 7 8 3 1 2 10 6 5 4 9 d. 11 8 7 9 54

II. Pitch Class and Mod 12: Pitches that are one or more octaves apart are equiva-lent members of a single pitch class. Because an octave contains twelve semi-tones, pitch classes can be discussed using arithmetic modulo 12 (mod 12), in which any integer larger than 11 or smaller than 0 can be reduced to an integer from 0 to 11 inclusive.

1.

2.

Using mod 12 arithmetic, reduce each of the following integers to an integer from 0 to 11 inclusive: a. 15 b. 27 c. 49 d. 13 e. -3 f. -10 g. -15

List at least three integers that are equivalent (mod 12) to each of the fol-lowing integers: a. 5 b. 7 c. 11

Basic Concepts and Definitions 17 3. Perform the following additions (mod 12):

a. 6+6 b. 9 + 10 c. 4+9 d. 7 + 8

4. Perform the following subtractions (mod 12): a. 9 10 b. 7- II c. 2-10 d. 3 8

III. Intervals: Intervals are identified by the number of semitones they contain.

IV.

1.

2.

For each of the following traditional interval names, give the number of semi tones in the interval: a. major third b. perfect fifth c. augmented sixth d. diminished seventh e. minor ninth f. major tenth

For each of the following numbers of semi tones, give at least one tradi-tional interval name: a. 4 b. 6 c. 9 d. 11 e. 15 f. 24

Ordered Pitch Intervals: A pitch interval is the interval between two pitches, counted in semi tones. + indicates an ascending interval; indicates a descend-ing interval.

1. Construct the following ordered pitch intervals on a musical staff, using middle C as your starting point. a. +15 b. -7 c. -4 d. +23

18

v.

2.

Basic Concepts and Definitions For the following melodies, identify the ordered pitch interval formed by each pair of adjacent notes.

Unordered Pitch Intervals: An unordered pitch interval is simply the space between two pitches, without regard to the order (ascending or descending) of the pitches.

1. Construct the following unordered pitch intervals on a musical staff, using middle C as the lowest note.

2.

a. 15 b. 4 c. 7 d. 11 e. 23

For the melodies in Exercise IV /2, identify the unordered pitch interval formed by each pair of adjacent notes.

VI. Ordered Pitch-Class Intervals: A pitch-class interval is the interval between two pitch classes. On the pitch-class clockface, always count clockwise from the first pitch class to the second.

1.

2.

For each of the melodies in Exercise IV /2, identify the ordered pitch-class interval formed by each pair of adjacent notes. Which ordered pitch-class intervals are formed by the following ordered pitch intervals? a. +7 b. -7

Basic Concepts and Definitions 19 c. +11 d. +13 e. -1 f. -6

VII. Unordered Pitch-Class Intervals: An unordered pitch-class interval is the short-est distance between two pitch classes, regardless of the order in which they occur. To calculate an unordered pitch-class interval, take the shortest route from the first pitch class to the second, going either clockwise or counterclock-wise on the pitch-class clockface.

1. For each of the melodies in Exercise IV /2, identify the unordered pitch-class interval formed by each pair of adjacent notes.

2. An unordered pitch-class interval is also called an interval class. Give at least three pitch intervals belonging to each of the six interval classes.

VIII. Interval-class Vector: Any sonority can be classified by the intervals it contains.

I.

The interval content is usually shown as a string of six numbers called an interval-class vector. The first number in the interval-class vector gives the number of occurrences of interval class I ; the second number gives the number of occur-rences of interval class 2; and so on.

1.

2.

For each of the following collections of notes, give the interval-class content, expressed as an interval-class vector. a. 0, 1, 3, 4, 6, 7, 9, 10 b. 0, 2, 4, 6, 8, 10 c. 2, 3, 7 d. the augmented triad e. the pentatonic scale f. 1, 5, 8, 9

For each of the following interval-class vectors, try to construct the col-lection that it represents. a. 111000 b. 004002 c. llllll d. 303630

ANALYSIS

Webern, Symphony Op. 21, Thema, mm. 1-11, clarinet melody: How is this melody organized? What patterns of recurrence do you notice? Begin by iden-tifying all of the ordered and unordered pitch and pitch-class intervals. (Hint: Consider not only the intervals formed between adjacent notes of the melody, but also the intervals that frame it, for example, the interval between the first note and the last, between the second note and the second-to-last, and so on.)

20 Basic Concepts and Definitions II. Schoenberg, Piano Concerto, mm. 1-8, right-hand melody: Are there any inter-

vals or motives that recur? (Hint: The melody is framed by its first note, B, which also is its highest note, its lowest note, AJ,, and its final note, G. Are there varied repetitions of this three-note motive directly within the melody?)

III. Stravinsky, "Musick to heare" from Three Shakespeare Songs, mm. 1-8, flute melody: What patterns of intervallic recurrence do you see? (Hint: Think of the first four notes as a basic motivic/intervallic structure.)

IV. Crawford, Diaphonic Suite No. I for Oboe or Flute, mm. 1-18: How is this melody organized? The composer thought of this melody as a kind of musical poem and indicated the lines of the poem with double bars (at the end ofmm. 5, 9, 14, and 18). Describe the musical "rhymes" and any other intervallic or motivic recurrences. (Hint: Take the first three notes, ip +2 followed by ip -1, as a basic motive.)

V. Varese, Octandre, mm. 1-5, oboe melody: How is this melody organized? (Hint: Consider both the first four notes, G>-F-E-D#, and the highest three of those, 01-E-DI, as basic motivic units.)

VI. Babbitt, "The Widow's Lament in Springtime," mm. 1-6, vocal melody: How is this melody organized intervallically and motivically? How many different ordered pitch-class intervals are used? Amid this variety, what are the sources of unity? (Hint: Consider the framing intervals-first to last, second to second-to-last, etc., as well as the direct melodic intervals.)

EAR-TRAINING AND MUSICIANSHIP

I. Webern, Symphony Op. 21, Thema: Sing the clarinet melody, accurately and in tempo, using pitch-class integers in place of the traditional solfege syllables. To maintain a single syllable for each note, sing "oh" for 0, "sev" for 7, and ''lev" for 11.

II. Schoenberg, Piano Concerto, mm. 1-8, right-hand melody: Sing the melody, accurately and in tempo, using pitch-class integers.

III. Stravinsky, "Musick to heare" from Three Shakespeare Songs, mm. 1-8, flute melody: Sing the melody, accurately and in tempo, using pitch-class integers.

IV. Crawford, Diaphonic Suite No. 1 for Oboe or Flute, mm. 1-18: Play this melody, accurately and in tempo, on any appropriate instrument.

V. Varese, Octandre, mm. 1-5, oboe melody: Play this melody, accurately and in tempo, on any appropriate instrument.

VI. Babbitt, "The Widow's Lament in Springtime," mm. 1-6, vocal melody: Sing the melody, accurately and in tempo, using either the words of the text (by William Carlos Williams) or pitch-class integers.

VII. Identify melodic and harmonic intervals played by your instructor as ordered and unordered pitch and pitch-class intervals.

VIII. From a given note, learn to sing above or below by a specified pitch interval (within the constraints of your vocal range).


COMPOSITION

I. Write two short melodies of contrasting character for solo flute or oboe that make extensive use of one of the following motives: ip, ip, i, or i.

II. Write brief duets for soprano and alto that have the following characteristics:

1. Begin with middle C in the alto and the B eleven semitones above it in the soprano.

2. Use whole notes only, as in first-species counterpoint. 3. The interval between the parts must be a member of ic l, ic2, or ic6. 4. Each part may move up or down only by ip1, ip2, ip3, or ip4. 5. End on the notes you began with. 6. Try to give an attractive, purposeful shape to both melodies.

Analysis 1

Webern, "Wie bin ich frob!" from Three Songs, Op. 25 Schoenberg, "Nacht:' from Pierrot Lunaire, Op. 21

Listen several times to a recording of "Wie bin ich froh!"-a song written by Anton Webern in 1935. We will concentrate on the first five measures, shown in Example Al-l.

Example Al-l Webern, "Wie bin ich froh!" from Three Songs, Op. 25 (mm. 1-5).

22

Analysis 1 Here is a translation of the first part of the text, a poem by Hildegarde Jone.

Wie bin ich froh! noch einmal wird mir alles griln und leuchtet so!

How happy I am! Once more all grows green around me And shines so!

The music may sound at first like disconnected blips of pitch and timbre. A texture that sounds fragmented, that shimmers with hard, bright colors, is typical ofWebern. Such a texture is sometimes called "pointillistic," after the technique of painting with sharply defined dots or points of paint. Gradually, with familiarity and with some knowledge of pitch and pitch-class intervals, the sense of each musical fragment and the interrelations among the fragments will come into focus.

The lack of a steady meter may initially contribute to the listener's disorientation. The notated meters, 3/4 and 4/4, are hard to discern by ear, since there is no regular pattern of strong and weak beats. The shifting tempo-there are three ritards in this short passage--confuses matters further. The music ebbs and flows rhythmically rather than following some strict pattern. Instead of searching for a regular meter, which certainly does not exist here, let's focus instead on the smaller rhythmic fig-ures in the piano part, and the ways they group to form larger rhythmic shapes.

The piano part begins with a rhythmic gesture consisting of three brief figures: a six-teenth-note triplet, a pair of eighth-notes, and a four-note chord. Except for two iso-lated single tones, the entire piano part uses only these three rhythmic figures. But, except for measure 2, the three figures never again occur in the same order or with the same amount of space between them. The subsequent music pulls apart, plays with, and reassembles the opening figures. Consider the placement of the sixteenth-note triplet, which becomes progressively more isolated as the passage progresses. In the pickup to measure I and in measure 2, it is followed immediately by the pair of eighth-notes. In measure 3, it is followed immediately, not by a pair of eighth-notes, but by a single note. At the beginning of measure 4, it is again followed by a single note, but only after an eighth-triplet rest. At the end of measure 4, it is even more iso-lated-it is followed by a long silence. The shifting placement of the rhythmic fig-ures gives a gently syncopated feeling to the piano part. You can sense this best if you play the piano part or tap out its rhythms.

Now let's turn to the melodic line. Begin by learning to sing it smoothly and accu-rately. This is made more difficult by the wide skips and disjunct contour so typical of Webern's melodic lines. Singing the line will become easier once its organization is better understood. Using the concepts of pitch and pitch class, and of pitch and pitch-class intervals, we can begin to understand how the melody is put together.

There is no way of knowing, in advance, which intervals or groups of intervals will turn out to be important in organizing this, or any, post-tonal work. Each of the post-tonal pieces discussed in this book tends to create and inhabit its own musical world, with musical content and modes of progression that may be, to a significant extent,

23

Analysis 1 independent of other pieces. As a result, each time we approach a new piece, we will have to pull ourselves up by our analytical bootstraps. The process is going to be one of trial and error. We will look, initially, for recurrences (of notes and intervals) and patterns of recurrence. It often works well to start right at the beginning, to see the ways in which the initial musical ideas echo throughout the line.

In "Wie bin ich froh," it turns out that the first three notes, G-E-D~, and the intervals they describe play a particularly central role in shaping the melody. Let's begin by considering their ordered pitch intervals (see Figure Al-l).

-3 +11 nn

G E D~ ~

+8

Figure Al-l The same ordered pitch intervals occur in the voice in two other places, in measure 3 (D-B-Bb) and again in measure 4 (C-A-GI). (See Example Al-2.)

ExampleAl-2 Three fragments with the same ordered pitch intervals. Sing these three fragments, then sing the whole melody and listen to how these frag-ments help give it shape. The second fragment starts five semi tones lower than the first, while the third fragment starts five semitones higher. That gives a sense of symmetry and balance to the melody, with the initial fragment lying halfway between its two direct rep-etitions. Furthermore, the second fragment brings in the lowest note of the melody, B, while the third fragment brings in the highest note, Gi. These notes, together with the initial G, create a distinctive frame for the melody as a whole, one which repli-cates the ordered pitch-class intervals of the initial fragment (see Example A 1-3).

24

11

Example Al-3 A melodic frame (first note, lowest note, highest note) that repli-cates the ordered pitch-class intervals of the initial fragment.

Analysis 1 Composers of post-tonal music often find ways of projecting a musical idea simulta-neously on the musical surface and over larger musical spans. This kind of composing-out is an important unifying device and it is one to which we will often return.

The three melody notes at the beginning of measure 3, 0-F-D, also relate to the opening three-note figure, but in a more subtle way. They use the same pitch intervals as the first three notes of the melody (3, 8, and 11 ), but the intervals occur in a differ-ent order. In addition, two of the three intervals have changed direction (see Figure Al-2).

-3 +11 nn G~D#

+8

FigureAl-2

In other words, the fragment 0-F-D has the same unordered pitch intervals as the opening figure, G-E-Di. The relationship is not as obvious as the one shown in Example Al-2, but it is still not hard to hear. Sing the two fragments, then sing the entire melody and listen for the resemblance (see Example Al-4).

Example A 1-4 1\vo fragments with the same unordered pitch intervals.

The first four pitch classes of the melody are the same, and in the same order, as the last four: G-E-DI-Fi (see ExampleAl-5).

ordered pitch intervals: -3 + 11 + 3 +9 -13 +3

4;;!Ef ] #] eJ n

ordered Wie bin ich froh! und leuch tet so! pitch-class mtervals: 9 11 3 9 11

Example Al-5 The first four notes and the last four have the same ordered pitch-class intervals. .

25

Analysis 1 The contours of the two phrases (their successive ordered pitch intervals) are differ-ent, but the ordered pitch-class intervals are the same: 9-11-3. This similarity between the beginning and end of the melody is a nice way of rounding off the melodic phrase and of reinforcing the rhyme in the text: "Wie bin ich frob! ... und leuchtet so!" Sing these two fragments and listen for the intervallic equivalence that lies beneath the change in contour. By changing the contour the second time around, Webern makes something interest-ing happen. He puts the E up in a high register, while keeping the G, Dl, and H together in a low register. Consider the unordered pitch-class intervals in that regis-trally defined three-note collection (G-m-Fi). It contains interval classes 1 (G-Fi), 3 (DI-Fi), and 4 (G-D~). These are exactly the same as those formed by the first three notes (G-E-D~) of the figure: E-DI is l, G-E is 3, and G-DI is 4 (see Example Al-6).

Example A 1-6 A registral grouping and a melodic figure contain the same unordered pitch-class intervals.

The melodic line is thus supercharged with a single basic motive. The entire melody develops musical ideas presented in the opening figure, sometimes by imitating its ordered pitch intervals, sometimes by imitating only its unordered pitch intervals, and sometimes, still more subtly, by imitating its ordered or unordered pitch-class intervals (see Example A 1-7).

ordered pitch-clasl> intervab

uoordered pilch intervals

ordered pitch intervals

unordered pitch-class imervals

ExampleAl-7 Development of the initial melodic figure.

Knowledge of the intervallic structure of the melody should make it easier to hear it clearly and to sing it accurately. Sing the melody again, concentrating on the motivic and intervallic interplay shown in Example A 1-7.

26

Analysis 1 The piano accompaniment develops and reinforces the same musical ideas. Rather than trying to deal with every note, let's just concentrate on the sixteenth-note triplet figure that comes five times in the passage. When it occurs in measure 2 (G-E-D!), it contains the same pitches and thus the same ordered pitch intervals as the beginning of the melody: + 11. In measure 3, different pitches are used (C-A-G~), but the ordered pitch intervals are the same: -3, + 11. When it occurs in the pickup to mea-sure I (FI-F-D) and at the end of measure 4 (B-B~G), it has the same ordered pitch intervals, but reversed: + 11, -3.

The remaining occurrence of the figure, at the beginning of measure 4 (C-A-0), is somewhat different from these. Its ordered pitch intervals are + 16. It is not com-parable to the others in terms of its pitch intervals or even its ordered pitch-class intervals. To understand its relationship to the other figures we will have to consider its interval classes. It contains a 3 (C-A), a 1 (C-0), and a 4 (A-CO. Its interval-class content (the interval-class vector is 101100) is thus the same as the first three notes of the voice melody (see Example Al-8).

Wie bin ich

Example A 1-8 Accompanimental figures derived from the initial melodic idea.

In fact, all of the three-note figures we have discussed in both the vocal and piano parts have this interval-class content. That is one reason the piece sounds so unified. Play each of the three-note figures in the piano part and listen for the ways they echo the beginning of the voice part-sometimes overtly, sometimes more subtly.

So far, we have talked about the voice part and the piano part separately. But, as in more traditional songs, the piano part both makes sense on its own and accompanies and supports the voice. For a brief example, consider the two single notes in the piano part, the F~ in measure 3 and the m in measure 4. In both cases, the piano note, together with nearby notes in the voice, creates a three-note collection with that familiar interval-class content: 10 II 00 (see Example A 1-9). The passage, at least as far as we have discussed it, is remarkably unified intervalli-

~ally.lt focuses intensively on the pitch intervals 3, 8, and II and, more abstractly, on mterval classes I, 3, and 4. The passage is saturated with these intervals and with motivic shapes created from them. Some of the relationships are simple and direct-we can discuss them in terms of shared pitch intervals. Others are subtly concealed

27

58 Pitch-Class Sets

1. Put the set into nonnal form. (Let's take [1,5,6,7] as an example.) 2. Transpose the set so that the first element is 0. (If we transpose [ 1 ,5,6,7] by T 11 ,

we get [0,4,5,6].) 3. Invert the set and repeat steps land 2. ([1,5,6,7] inverts to {11,7,6,5]. The nor-

mal fonn of that set is [5,6,7,11]. If that set is transposed at T7, we get [0,1,2,6].)

4. Compare the results of step 2 and step 3; whichever is more packed to the left is the prime form. ([0,1,2,6] is more packed to the left than [0,4,5,6], so (0126) is the prime form of the set class of which [1,5,6,7], our example, is a member.)

As with normal form, it will often be possible to detennine the prime fonn just by inspecting a set displayed around a pitch-class clockface. Find the widest gap between the pitch classes. Assign zero to the note at the end of the gap and read off a possible prime form clockwise. Then assign zero to the note at the beginning of the gap and read off another possible prime form counterclockwise. (If there are two gaps of the same size, choose the one that has another relatively big gap right next to it.) Whichever of these potential prime fonns has fewer big integers is the true prime fonn. Figure 2-20 illustrates with the four sets we used in Example 2-5 to determine nor-mal form. It is basically a matter of visualization, and it will get easier with practice.

4~~h It

E

SFI F

sc (0237) sc (0147) sc (0148)

Figure2-20

In Appendix 1, you will find a list of set classes showing the prime fonn of each. If you think you have put a set in prime fonn but you can't find it on the list, you have done something wrong. Notice, in Appendix 1, how few prime fonns (set classes) there are. With our twelve pitch classes, it is possible to construct 220 differ-ent trichords (three-member sets). However, these different trichords can be grouped into just twelve different trichordal set classes. Similarly, there are only twenty-nine tetrachord classes (four-member sets), thirty-eight pentachord classes (five-member sets), and fifty hexachord classes (six-member sets). We will defer discussion of sets with more than six elements until later.

The list of set classes in Appendix 1 is constructed so as to make a great deal of useful information readily available. Any sonority of between three and nine ele-ments is a member of one of the set classes listed here. In the first column, you will see a list of prime forms, arranged in ascending order. The second column gives

Pitch-Class Sets 59

Forte's name for each set class. The third column contains the interval-class vector for the set class. (This is the interval-class vector for every member of the set class. since interval content is not changed by transposition or inversion.) In the fourth col~ umn are two numbers separated by a comma; these numbers measure the transposi-tional and inversional symmetry of the set class-we will discuss these concepts later. Across from each trichord, tetrachord, and pentachord, and some of the hexa-chords, is another set with all of its relevant infonnation in the reverse order. We will discuss these larger sets later.

Segmentation and Analysis In the post-tonal music discussed in this book, coherence is often created by relation-ships among sets within a set class. It is possible to hear pathways through the music as one or more sets are transposed and inverted in purposeful, directed ways. Often, we find that there is not one single best way to hear our way through a piece; rather, our hearings often need to be multiple, as the different paths intersect, diverge, or run parallel to each other. To use a different metaphor, post-tonal music is often like a rich and varied fabric, comprised of many different strands. As we try to comprehend the music, it is our task to tease out the strands for inspection, and then to see how they combine to create the larger fabric.

One of our main analytical tasks, then, is to find the principal sets and show how they are transposed and inverted. But how do you know which sets are the important ones? The answer is that you cannot know in advance. You have to enter the world of the piece-listening, playing, and singing-until you get a sense of which musical ideas are fundamental and recurring. In the process, you will find yourself moving around a familiar kind of conceptual circle. You can't know what the main ideas are until you see them recur; but you can't find recurrences until you know what the main ideas are. The only practical solution is to poke around in the piece, proposing and testing hypotheses as you go. In the process, you will be consid-ering many different segmentations of the music, that is, ways of carving it up into meaningful musical groupings.

When you have identified what you think may be a significant musical idea, then look carefully, thoroughly, and imaginatively for its transposed or inverted recurrences. Here are some places to look (this list is not exhaustive!):

1.

2.

In a melodic line, consider all of the melodic segments. For example, if the melody is six notes long, then notes 1-2-3, 2-3-4, 3-4-5, and 4-5-6 are all viable three-note groupings. Some of these groupings may span across rests or phrasing boundaries, and that is okay. A rich interaction between phrase struc-ture and set-class structure is a familiar feature of post-tonal music. Hannonically, don't restrict yourself just to chords where all of the notes are attacked at the same time. Rather, consider all of the simultaneities, that is, the notes sounding simultaneously at any particular point. Move through the music like a cursor across a page, considering all of the notes sounding at each moment.

60 Pitch-Class Sets

3.

4.

5.

Notes can be associated by register. In a melody or a phrase, consider the high-est (or lowest) notes, or the high points (or low points) of success~ve phrases. Notes can be associated rhythmically in a variety of ways. Constder as a pos-sible group the notes heard on successive downbeats, or the n~tes heard at the beginning of a recurring rhythmic figure, or the notes that are gtven the longest durations. Notes can be associated timbrally in a variety of ways. Consider as a pos.sible group notes that are produced in some dis.tinc~ive way, ~or e~ample, by a smgle instrument in an ensemble, or by a certam kind of arttculatwn (e.g., staccato, pizzicato).

In all of your musical segmentations, strive for a balance bet~een imaginative see~ing and musical common sense. On the one hand, do not restnct yourself to the obvt-ous groupings (although these are often a good place to start). Interesting relationships may not be apparent the first, or second, or third time through, and you need to be thorough and persistent in your investigations. On the other hand, you have to stay within the boundaries of what can be meaningfully heard. You .can't pluck notes out in some random way, just because they form a set t~at you are mte.r-ested in. Rather, the notes you group together must be associated wtth each other m some musical way. They have to share some distinctive quality (for example, of prox-imity, or highness, or lowness, or loudness, or longness) that groups them together and distinguishes them from the other notes around them. If, after ~orne repeated, good-faith effort to hear a certain musical grouping, Y?U cannot ma~e It palpa~ly real for yourself, then abandon it and go on to the next thmg. The g?alts ~o descnbe t~e richest possible network of musical relationships, to let our mustcal mt~ds an~ must-cal ears lead each other along the many enjoyable pathways through thts music.

In analyzing post-tonal music, you should feel free to draw on the full range of concepts developed in Chapters 1 and 2 of this book, from the most concrete to the most abstract (see Figure 2-21).

Pitch -+ Pitch-class Pitch intervals -+ Ordered pitch-class intervals -+ Interval classes Pitch-class set -+ Tn-type -+ Set class (Tnffnl-type)

Figure 2-21

In the process, you may discover how hard it is to find explanations fo: e:e~ single note in a piece or even a short passage. One familiar feature.of this mus1c .Is tts resistance to single, all-encompassing explanations. Instead of trymg. to fi~d a sm~le source for all of the music, try to forge meaningful networks of relatwnship, teasmg out particularly striking strands in the musical fabric, and following. a few inter.esting musical paths. That is an attainable and satisfying goal for musical analysts and musical hearing.

Pitch-Class Sets 61

BIBLIOGRAPHY

Many of the basic concepts of post-tonal theory have their origins in the writing and teaching of Milton Babbitt. On normal form, see "Set Structure as a Compositional Determinant" (1961), reprinted in The Collected Essays of Milton Babbitt, ed. Stepen Peles, Stephen Dembski, Andrew Mead, and Joseph Straus (Princeton: Princeton University Press, 2003), pp. 86-198. On index number, see "Twelve-Tone Rhythmic Structure and the Electronic Medium" ( 1962) and "Contemporary Music Composition and Music Theory as Contemporary Intellectual History" ( 1971 ), both reprinted in The Collected Essays of Milton Babbitt, pp. 109~140 and 270-307. It would be difficult to overestimate Babbitt's influence on post-tonal theory.

Allen Forte (The Structure of Atonal Music) and John Rahn (Basic Atonal Theory) pre-sent slightly different criteria for normal form and for prime form, but these result in only a small number of discrepancies, affecting the following set classes: 5-20, 6-29, 6-31,7-18,7-20, and 8-26. This book adopts Rahn's formulation.

The x;model of inversion is David Lewin's. See his Generalized Musical Intervals and Tran.iformations (New Haven: Yale University Press, 1987), pp. 50-56.

Schoenberg's Piano Piece, Op. II, No. I. has been widely analyzed. George Perle dis-cusses its intensive use of set class 3-3 (014) (which he calls a "basic cell") in Serial Composition and Atonality. See also Allen Forte, "The Magical Kaleidoscope: Schoenberg's First Atonal Masterwork, Opus II, No. 1," Journal of the Arnold Schoenberg Institute 5 ( 1981 ), pp. 127-68; and Gary Wittlich, "Intervallic Set Structure in Schoenberg's Op. 11, No.1," Perspectives of New Music 13 (1974), pp. 41-55. Ethan Haimo uses the work as a starting point for a critique of pitch-class set theory in "Atonality, Analysis, and the Intentional Fallacy," Music Theory Spectrum 18/2 ( 1996), pp. 167-99. See also a refined network analysis by David Lewin in "Some Aspects of Voice Leading Between Pcsets," Journal of Music Theory 42/1 ( 1998), pp. 15-72.

The problems of segmentation and musical grouping are discussed in Christopher Hasty, "Segmentation and Process in Post-Tonal Music;' Music Theory Spectrum 3 ( 1981 ), pp. 54-73 and Dora Hanninen, "Orientations, Criteria, Segments: A General Theory of Segmentation for Music Analysis," Journal of Music Theory 4512 (200 1 ), pp. 345-434.

Exercises

I.

THEORY

Normal Form: The normal form of a pitch-class set is its most compact repre-sentation.

1. Put the following collections into normal form on a musical staff.

62 Pitch-Class Sets

2. Put the following collections into normal form using integers. Write your answer within square brackets. a. 11, 5, 7, 2 b. 0, 10,5 c. 7, 6, 9, 1 d. 4, 7,2, 7,11 e. the C-major scale f. C,B,m,E,G g. 9, 11, 2, 5, 9, 8, 1, 2

11. Transposition: Transposition (Tn) involves adding some transposition interval (n) to each member of a pitch-class set. Two pitch-class sets are related by T n if, for each element in the first set, there is a corresponding element in the second set n semitones away.

1. Transpose the following pitch-class sets as indicated. The sets are given in normal form; be sure your answer is in normal form. Write your answer on a musical staff.

2. Transpose the following pitch-class sets as indicated. Write your answers in normal form using integer notation. a. T3 [8,0,3] b. T9 [1,4,7,10] c. T6 [5,7,9,11,2] d. T7 [9,11,1,2,4,6]

3. Are the following pairs of pitch-class sets related by transposition? If so, what is the interval of transposition? All the sets are given in normal form. a. [8,9,11,0,4] [4,5,7,8,0] b. [7,9,1] [1,5,7] c. [7,8,10,1,4] [1,2,4,7,10] d. [1,2,5,9] [11,0,3,7]

III. Inversion: Inversion (Tnl) involves inverting each member of a pitch-class set (subtracting it from 12), then transposing by some interval n (which may be 0). Two sets are related by inversion if they can be written so that the interval suc-cession of one is the reverse of the interval succession of the other.

IV.

v.

Pitch-Class Sets 63

1. Invert the following pitch-class sets as indicated. Put your answer in nor-mal form and write it on a musical staff.

2. Invert the following pitch-class sets as indicated. Use integer notation and put your answer in normal form. a. T9I [9,10,0,2] b. T 0I [1 ,2,5] c. T~I [1,2,4,7,10] d. T 10I [ 10,11 ,0,3,4,7] e. T6I [4,7,10,0] f. T4I (the C-major scale)

3. Are the following pairs of pitch-class sets related by inversion? If so, what is value of n in T nl? All sets are given in normal form. a. [2,4,5,7] [8,10,11,1] b. [4,6,9] [4,7,9] c. [1,2,6,8] [9, 11,2,3] d. [4,5,6,8,10,1] [6,8,10,11,0,3] e. [8,9,0,4] [8,11,0,4]

Index Number: In sets related by inversion (T nl), the corresponding elements sum to n. When the sets are in normal form, the first element of one usually corresponds to the last element of the other, the second element of one corre-sponds to the second-to-last element of the other, and so on.

1.

2.

For each of the following pairs of inversionally related sets, figure out the index number. Sets are given in normal form. a. [5,9,11] [7,9,1] b. [ 4,5,8, 11] [ 10, I ,4,5] c. [4,5,8,0] [9,0,1,5] d. [1,3,6,9] [10,1,4,6]

Using ~o~r knowledge of index numbers, invert each of the following sets as mdtcated. Put your answer in normal form. a. T3I [1,3,5,8] b. T9I [10,1,3,6] c. TOI [1,2,4,6,9] d. T41 [4,5,6,7]

Inversion: Inversion (19) involves mapping each note in a pitch-class set onto a corresponding note by performing whatever inversion maps x onto y.

64

VI.

1.

2.

3.

Pitch-Class Sets

Invert the following pitch-class set as indicated. Put your answer in nor-mal form and write it on a musical staff.

d. e.

Invert the following pitch-class sets as indicated. Put your answer in nor-mal form. a. ~~~ (G, AI>, B~. B] b. r'j,"[B, C, D, F, Fl] c. Ig [B, C, D, E, F, G] d. I~!(FI, G~. B, 0] e. I~ [0, Dl, G, A] Using the Jll notation, give at least two labels for the operation that con-nects the following pairs of inversionally related sets. a. [G, Gl, B] [G, m, B] b. [CI, D, F, G][G, A, C, 0] c. [A~, A. D~. B] [A, B, Dl, E] d. [D, F, A] [F, A, C] e. [G!, A. A~. B, C, D) [0, m, E, F, Fl, G]

Prime Form: The prime form is the way of writing a set that is most compact and most packed to the left, and begins on 0. 1. Put each of the following pitch-class sets in prime form. All sets are

2.

given in normal form. a. [10,3,4] b. [7,8,11,0,1,3] c. [G,B,D] d. [2,5,8,10] e. [4,6,9,10,1] f. [O,D,G,A~]

Are the following pitch-class sets in prime form? If not, put them in prime form. a. (0, 1,7) b. (0,2,8) c. (0,2,6,9) d. (0,1,4,5,8,9)

VII. The List of Set Classes (Appendix 1) l. Name all the tetrachords that contain two tritones. b h d? 2. What is the largest number of interval class 4s contained y a tetrac or .

Which tetrachords contain that many?

I.

II.

III.

IV.

v.

I.

3. 4.

5.

6.

Pitch-Class Sets 65

Which trichord(s) contain both a semitone and a tritone? Which tetrachords contain one occurrence of each interval class? (Notice that they have different prime forms.) How many trichords are there? How many nonachords (nine-note sets)? Why are these numbers the same? Which hexachords have no occurrences of some interval? of more than one interval?

7. Which hexachord(s) have the maximum (six) occurrences of some inter-val? Which have five occurrences of some interval?

8. Are there any sets that contain only one kind of interval?

ANALYSIS

Crawford, Piano Prelude No. 9, mm. 1-9. (Hint: Begin by considering the upper and lower parts separately, but consider also the harmonies formed between them.) Webem, Concerto for Nine Instruments, Op. 24, second movement, mm. 1-11. (Hint: Begin by considering the first three melody notes (G-DI-E) as a basic motive and relate it to transpositionally equivalent repetitions of it. Then con-sider the three notes in measure 1 (G-Bl>-B) as a basic motive, and relate it to its transposed repetitions. Finally, combine those two transpositional paths into a single comprehensive view.) Stravinsky, Agon, mm. 418 (with pick-up)-429. (Hint: Begin with the caden-tial chord, [B:,, B, D~. D] and show how it relates to the music that precedes it.) Webem, Movements for String Quartet, Op. 5, No. 2, beginning through the downbeat of m. 4. (Hint: Begin by treating the first three notes in the viola, G-B-0, as a basic motivic unit. Look for transposed and inverted repetitions.) Babbitt, Semi-Simple Variations, Theme, mm. 1-6. (Hint: Imagine the passage as consisting of four registral lines, a soprano that begins on m, an alto that begins on D, a tenor that begins on A, and a bass that begins on Cl. Each line consists of six different notes. Analyze the lines separately [with particular attention to-their trichords] and in relation to each other.)

EAR-TRAINING AND MUSICIANSHIP

Crawford, Piano Prelude No. 9, mm. 1-9. Pianists: play the entire passage. Nonpianists: play the treble parts only. It's mostly a duet-use one hand for each line.

II. Webern, Concerto for Nine Instruments, Op. 24, second movement, mm. I-ll. The passage (and, indeed, the entire movement) is divided into a melody, shared by eight melodic instruments, and a piano accompaniment. Learn to play melody and accompaniment separately, then together. Learn to sing the

66

III.

IV.

v.

VI.

Pitch-Class Sets

melody, using pitch-class integers in the place of solfege syllables (you will have to transpose all or part of the melody into a comfortable register). Stravinsky, Agon, mm. 418 (with pick-up)-429. Play this passage accurately and in tempo at the piano (the tempo is Adagio, ~'l= 112). Webern, Movements for String Quartet, Op. 5, No. 2, beginning through the downbeat of m. 4. Sing the viola melody using pitch-class integers while play-ing the accompanying chords on the piano. Babbitt, Semi-Simple Variations, Theme, mm. 1-6. Write out each of the regis-tral lines as six consecutive whole notes, then learn to sing each one smoothly and accurately. Learn to identify the twelve different trichordal set-classes when they are played by your instructor. It may be easier if you learn them in the following order, adding each new one as the previous ones are mastered:

1. 3-1 (012): chromatic trichord 2. 3-9 (027): stack of perfect fourths or fifths 3. 3-ll (037): major or minor triad 4. 3-3 (014): major and minor third combined 5. 3-7 (025): diatonic trichord 6. 3-12 (048): augmented triad 7. 3-5 (016): semitone and tritone 8. 3-8 (026): whole-tone and tritone 9. 3-10 (036): diminished triad

10. 3-2 (013): nearly chromatic 11. 3-6 (024): two whole-tones 12. 3-4 (015): semitone and perfect fourth

COMPOSITION

I. Take the first measure or two of one of the compositions discussed in the Analysis section, and, without looking ahead, continue and conclude your own brief composition. Then compare your composition with the published proto-type.

II. Write a short piece for your instrument in which the main sense of direction is provided by the purposeful, directed successive transposition of a pitch-class set of your choice.

Analysis 2 Schoenberg, Book of the Hanging Gardens,

Op. 15, No. 11 Bartok, String Quartet No. 4, first movement

Schoenberg wrote his Book of the Hanging Gardens, Op. 15, in 1908. The song cycle contains fifteen musical settings of poems by Stefan George. The "hanging gardens" described in the poems are those of ancient Babylon, one of the wonders of the ancient world. The gardens appear in George's poems as a kind of magical, ambigu-ous background for disturbing and inconclusive erotic verse. We will concentrate on the eleventh song in the cycle, but you should be familiar with the others as welL The music for the first thirteen measures of the song, the focus of our discussion, can be found in Example A2-l.

' Sehrruhig (.lc- m)

(continued)

Example A2-l Schoenberg, Book o.fthe Hanging Gardens, Op. 15, No. 11 (mm. 1-13).

67

Analysis 2 ( sehrruhig)

ten. war - - den un. er- dach -

Example A2-l (continued)

Als wir hinter dem bebliimten Tore Endlich nur das eigne Hauchen spiirten Warden uns erdachte Seligkeiten? Ich erinnere ...

When we, behind the flowering gate, At last felt only our own breathing, Was our bliss only imagined? I remember ...

Learn to sing the melodic line and to play the piano part (neither is difficult). Better yet, learn to sing the melodic line while accompanying yourself.

Let's begin by concentrating on the opening melodic gesture in the right hand of the piano part (see Example A2-2).

Example A2-2 The opening melodic gesture and its components.

68

Analysis 2 The four-note gesture is a member of set class 4-17 (0347). It is easy to visualize this melody as a triad with both major and minor thirds, although, as we will see, it occurs later in the song in a variety of guises. The gesture also contains three smaller musi-cal ideas that will become important later: the rising minor triad with which it begins (set class 3-11 (037)), the ascending interval of seven semitones spanned by that triad and divided into a +3 followed by a +4, and the final three notes of the gesture (set class 3-3 (014)). This last set class also is formed by the lowest three notes of the melody, BI>-DI>-D. Sing or play the figure and listen until you can hear all of these musical ideas. Then let us see how this melodic gesture and its components get devel-oped in the subsequent music.

In mea.;;ure 13, at the end of the passage we are considering, the same gesture comes back at the original transposition level in the piano, and almost simultaneously at T2 in the voice. These direct references to the opening are particularly appropriate to the text, since the singer at this moment is saying, "Ich erinnere" ("I remember"). The music conveys a sense of memory by recalling musical events heard earlier.

The vocal line beginning in measure 8 also contains echoes, slightly more concealed, of the same melodic gesture (see ExampleA2-3).

Als wir hin ter dem be- bliim - ten To re end lich nur das eig -

ne Hau chen sptir ten.

Example A2-3 Some statements of 4-17 (0347) in the vocal line.

The singer begins with a varied statement of the gesture at T10 (The B~ is an added passing note.) Shortly thereafter, it repeats T10 (again in a varied order), and the phrase concludes with T2 At that moment (the end of measure 10), chords are heard again in the piano, closing off the phrase. Melodically, the voice has used forms of 4-17 (0347) at both two semitones above (T2) and two semitones below (T10) the original form. These two transposition levels, one a little higher and one a little lower than the original one, give a sense of "not-quite-right"ness to the music that perhaps reflects the uncertainty in the upcoming text: "Was our bliss only imagined?" The singer would like to get back to the form that begins on B~, but she hasn't quite gotten there yet. Sing the vocal line again, and listen in it for these slightly off-center echoes of the opening melodic gesture. That gesture is developed in an even more concealed

69

Analysis 2 way in the piano part in measures 3 and 4. The right-hand part in those measures con-tains two new forms of4-17 (0347), atT3 and T7 (see ExampleA2-4).

Example A2-4 Statements of 4-17 (0347) in the piano introduction.

Play the piano part in those measures and listen for the resemblances to the opening gesture. Notice how the intervals from that opening gesture are rearranged within the chords. In the T7 version, for example, notice that the melody, C-A-G#, is the same set class as the last three notes in the opening gesture: 3-3 (014). Now let us see if we can assemble the five forms of 4-1

straus introtoposttonaltheory

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